Determine whether each function has an inverse function. if it does, find the inverse function and state any restrictions on its domain.

1.

2.

3.

QUESTION 1

The given function is;



This is an absolute value function.

For this function to have an inverse it must be a one-to-one function.

This absolute value function is not one-to-one
because


and



Since this function has more than one x-value mapping onto one y-value, it is not one-to-one and cannot have an inverse.

You can see from the graph that this function cannot pass the horizontal line test.

QUESTION 2

The given function is



Let



This implies that,



We see clearly that, this function is a circle that is centered at the origin.

This means that,



is a semicircle.

This function will not pass the horizontal line test and hence does not have an inverse.

QUESTION 3

The given function is



If we put



We obtain,











This shows that h(x) has an inverse because it is one-to-one.

Let



We interchange x and y to get,







Group like terms to get,



Factor to get,


Solve for y,



Hence



where